Algorithms for the On-Line Quota Traveling Salesman Problem ⋆ G. Ausiello 1 , M. Demange 2 , L. Laura 1 , and V. Paschos 3 1 Dip. di Informatica e Sistemistica Universit` a di Roma ”La Sapienza” Via Salaria 113 00198 Roma Italy. {ausiello,laura}@dis.uniroma1.it 2 ESSEC demange@essec.fr 3 Universit´ e Paris-Dauphine, Place du Mar´ echal De Lattre de Tassigny, 75775 Paris Cedex 16, France paschos@lamsade.dauphine.fr Abstract. The Quota Traveling Salesman Problem is a generalization of the well known Traveling Salesman Problem. The goal of the traveling salesman is, in this case, to reach a given quota of the sales, minimizing the amount of time. In this paper we address the on-line version of the problem, where requests are given over time. We present algorithms for various metric spaces, and analyze their performance in the usual frame- work of the competitive analysis. In particular we present a 2-competitive algorithm that matches the lower bound for general metric spaces. In the case of the half-line metric space, we show that it is helpful not to move at full speed, and this approach is also used to derive the best on-line polynomial time algorithm known so far for the more general On-Line TSP problem (in the homing version). 1 Introduction Let us imagine that a traveling salesman is not forced to visit all cities in a single tour but in each city he can sell a certain amount of merchandise and his committment is to reach a given quota of sales, by visiting a sufficient number of cities; then he is allowed to return back home. The problem to minimize the amount of time in which the traveling salesman fulfills his commitment is known as the Quota Traveling Salesman Problem (QTSP for short, see [4,9] for a definition of the problem) and it is also called Quorum-Cast problem in [11]. Such problem can be seen as a special case of the Prize-Collecting Traveling Salesman Problem (PCTSP 4 , [6]) in which again the salesman has to fulfill a quota but now nonnegative penalties are associated to the cities and the cost of the salesman tour is the sum of the distance traveled and the penalties for the non visited cities. QTSP corresponds to the case of PCTSP in which all penalties ⋆ Work of Giorgio Ausiello and Luigi Laura is partially supported by the Future and Emerging Technologies programme of the EU under contract number IST-1999- 14186 (ALCOM-FT); and by “Progetto ALINWEB: Algoritmica per Internet e per il Web”, MIUR Programmi di Ricerca Scientifica di Rilevante Interesse Nazionale. 4 Note that some authors use to name PCTSP the special case that we have called QTSP [5, 8].